Paper SDA-12 Stationarity Testing in High Frequency Seasonal Time Series

Time series quite often show patterns that repeat periodically. Monthly retail sales provide a good example. If the seasonality is very regular, seasonal dummy variables can be used to give, for example, a monthly effect for each month. With this approach, the January effect is assumed to be the same regardless of the year. Seasonal ARMA error terms can be added to make some local modifications. An alternate model that is useful when the seasonality changes over the years is the seasonal unit root model. Motivated by Box and Jenkins’ approach to modeling international airline ticket sales, this method takes a span d difference for seasonality d, e.g. d=12 for monthly data, and analyzes these seasonal span differences. Using the backshift operator B, the polynomial (1-B) represents the span d difference. Tables of percentiles for testing that the polynomial has unit roots (as does 1-B) are available (Dickey, Hasza, Fuller, 1984, henceforth “DHF”) for seasonal periods d=2, 4, and 12. As with ordinary (d=1) unit root tests, these are nonstandard distributions that shift when typical deterministic inputs like seasonal means are included in the model. It is possible that a user may want to test for unit roots at a longer lag, for example one might suspect periodicity 24 or 7x24=168 in hourly data and hence might ask if unit roots at those lags give an appropriate model. This paper deals with large d results for unit root tests. Some features emerge that are nicer than those of the shorter period cases. This paper is a slight modification of a paper (Dickey, 2008) delivered at the 2008 Joint Statistics Meetings. A followup paper with more mathematical detail and somewhat improved but more complex adjustments is to appear in the Journal of the Korean Statistical Society in 2010.